Class 11

Math

Algebra

Sequences and Series

If in an A.P. the sum of p terms is equal to sum of q terms, then prove that the sum of $p+q$ terms is zero.

$∴2P [2a+(p−1)d]=2q [2a+(q−1)d]$

or $(2a−d)(p−q)+(p_{2}−q_{2})d=0$,

cancel $p−q$ as $p=q$

or $2a−d+(p+q)d=0$

or $2a+(p+q−1)d=0$ .$(1)$

$∴S_{p+q}=2p+q [2a+(p+q−1)d]$

$=2p+q .0=0$, by $(1)$.